My husband I were married in Idaho but live in Washington. How do I check to see if the marriage was recorded in the state of WA

Unless you applied for a marriage license in WA and married in WA, there would normally not be any government office in WA that would know of your marriage outside of the state (unless you take steps to inform the government). WA does not require that spouses married outside of WA to report to any WA governmental office that the spouses are married.

Your employer or its insurance company may ask that you provide proof of your marriage. Generally, that proof can be a copy of your marriage license.